2017-10-10 21:34:52 UTC [313]: [1-1] user= db= host= FATAL: could not connect to the primary server: FATAL: no pg_hba.conf entry for replication connection from host "10.10.10.10", user "postgres", SSL off where "10.10.10.10" is my slave (slave.example.com) ip.
The comments and note under "address" at the following link suggest that what you are trying is possible. Hopefully they can shed some light as to what to check in order to find out why it is not working in your case. In particular, are both DNS entries configured?
Ensure you reload the configuration after making changes.
I don't recall whether having a all/all/any entry above your replication entry is going to be problematic - but that wouldn't explain a "no pg_hba.conf entry" error.